Mrs May did not disclose the name of the minister but told BBC Mersyside they did not want to meet grieving families and “it exemplified a sense” of the attitude of some colleagues at the time she was home secretary.
Other Tories believed the 1989 crush during the FA Cup semi-final, in which 97 Liverpool supporters died, happened “so long ago there’s no point in revisiting it”, Mrs May said.
But she said meeting the victims’ relatives helped convince her to pursue work which led to the High Court quashing the original inquest verdicts of “accidental death” in 2012.
Fresh inquests returned a conclusion of “unlawful killing” for those who died.
An independent panel had been set up by the previous Labour government and was to consider and publish documents relating to the disaster which had never been made public.
Mrs May had to decide whether or not to continue it as the new home secretary in 2010.
“I think it was a case that the families couldn’t trust anyone in authority and they’d been hitting their head against a brick wall for so long,” Mrs May told BBC Merseyside.
She said the victims’ families were “absolutely clear that the fans weren’t responsible for what had happened”.
“It shone through in everything they said to me,” she added.
The panel found that no Liverpool fans were responsible in any way for the disaster, and revealed that police statements had been changed in the aftermath seeking to blame fans.
“I have to confess that at the time I believed the stories that the authorities were saying - hearing all the stories that were being put out that it was the fault of the Liverpool fans,” Mrs May told BBC Merseyside.
She said there was a narrative at the time that focused on football hooligans and violence at matches and that fit into “what was being told by the authorities” and “people accepted it”.
Last month it was revealed updated Crown Prosecution Service (CPS) guidance is seeking to tackle tragedy-related abuse which is when fans sing, chant or gesture offensive messages about disasters or accidents involving players or fans, including the Hillsborough tragedy.
Douglas Mackay, of the CPS, said: “A small minority of so-called fans are both damaging the reputation of the sport and more importantly this offending has a devastating impact on the families of victims of tragedies and the communities connected closely to these events.”
In June Tottenham fan Kieron Darlow, 25, was banned from football matches for three years after being found guilty of mocking the Hillsborough disaster.